In the realm of mathematics, particularly within algebra, sequences play a pivotal role. They are essentially ordered lists of numbers that adhere to a specific pattern. This section delves into two primary types of sequences: Arithmetic Progressions (AP) and Geometric Progressions (GP). These sequences are integral for understanding numerical patterns and relationships.

**Arithmetic Progressions (AP)**

An Arithmetic Progression is a sequence where each term after the first is derived by adding a constant, known as the common difference, to the preceding term.

**Definition and Formula**

**General Form**: An AP is denoted as $a, a+d, a+2d, a+3d, \ldots$, where $a$represents the first term and $d$ is the common difference.**Nth Term Formula**: The nth term of an AP is given by $a_n = a + (n - 1)d$.**Sum Formula**: The sum of the first n terms of an AP is calculated as $S_n = \frac{n}{2}(2a + (n - 1)d).$

**Example 1: Finding the 10th Term of a Sequence**

Find the 10th term of the sequence $3, 7, 11, \ldots$.

**Solution:**

**1. Identify the First Term and Common Difference**

- First term $(a)$ = 3
- Common difference $(d)$ = 7 - 3 = 4

**2. Apply the Formula for the (n)th Term of an Arithmetic Progression (AP)**

- The $nth$ term of an AP is given by: $a_n = a + (n - 1)d$
- For the 10th term $(a{10}): a{10} = 3 + (10 - 1) \times 4$
- Calculate: $a_{10} = 3 + 9 \times 4 = 3 + 36 = 39$

**Answer:**

The 10th term of the sequence is **39**.

**Example 2: 20th Term and Sum of the First 20 Terms**

Determine the 20th term and the sum of the first 20 terms of the sequence $4, 9, 14, \ldots$

**Solution:**

**1. Identify the First Term and Common Difference:**

- First term $(a)$ = 4
- Common difference $(d)$ = 9 - 4 = 5

**2. Find the 20th Term:**

- Apply the $n$th term formula: $a_{20} = 4 + (20 - 1) \times 5$
- Calculate: $a_{20} = 4 + 19 \times 5 = 4 + 95 = 99$

**3. Calculate the Sum of the First 20 Terms:**

- The sum of the first $n$ terms of an AP is given by: $S_n = \frac{n}{2} [2a + (n - 1)d]$
- For the first 20 terms $(S_{20}):$
- Calculate: $S_{20} = 10 [8 + 19 \times 5] = 10 [8 + 95] = 10 \times 103 = 1030$

**Geometric Progressions (GP)**

A Geometric Progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

**Definition and Formula**

**General Form**: A GP is represented as $a, ar, ar^2, ar^3, \ldots$, where $a$ is the first term and $r$ is the common ratio.**Nth Term Formula**: The nth term of a GP is $a_n = ar^{n-1}$.**Sum Formula**: The sum of the first n terms of a GP is $S_n = a\frac{1 - r^n}{1 - r}$, for $r \neq 1$.

**Example 1: Sum of the First 5 Terms of a Geometric Sequence**

Find the sum of the first 5 terms of the sequence $2, 6, 18, 54, \ldots$

**Solution:**

**1. Identify the First Term and Common Ratio**

- First term $(a)$ = 2
- Common ratio $(r)$ = $\frac{6}{2} = 3$

**2. Apply the Formula for the Sum of the First **$n$** Terms of a Geometric Progression (GP)**

- The sum of the first $n$ terms of a GP is given by: $S_n = a \frac{1 - r^n}{1 - r}$
- For the first 5 terms $(S_5)$: $S_5 = 2 \frac{1 - 3^5}{1 - 3}$
- Calculate: $S_5 = 2 \frac{1 - 243}{1 - 3} = 2 \frac{-242}{-2} = 242$

**Answer:**

The sum of the first 5 terms of the sequence is **242**.

**Example 2: 7th Term and Sum of the First 7 Terms of a Geometric Sequence**

Determine the 7th term and the sum of the first 7 terms of the sequence ($5, 10, 20, 40, \ldots$

**Solution:**

**1. Identify the First Term and Common Ratio**

- First term $(a)$ = 5
- Common ratio $(r)$ = $\frac{10}{5}$ = 2

**2. Find the 7th Term**

- The $n$th term of a GP is given by: $a_n = a \times r^{n-1}$
- For the 7th term $(a_7)$: $a_7 = 5 \times 2^{7-1} = 5 \times 2^6$
- Calculate: $a_7 = 5 \times 64 = 320$

**3. Calculate the Sum of the First 7 Terms**

- The sum of the first $n$ terms of a GP is given by: $S_n = a \frac{1 - r^n}{1 - r}$
- For the first 7 terms $(S_7)$: $S_7 = 5 \frac{1 - 2^7}{1 - 2}$
- Calculate: $S_7 = 5 \frac{1 - 128}{-1} = 5 \times 127 = 635$

**Answer**

The seventh term of the sequence is **320** and the sum of the sequence is **635**.